#94335

order	512
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1353
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	C₂×#423	C₂²
socle	C₂	#7251
prime radical	C₂⁴	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂×#423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,11,13,27)(4,15)(5,19)(6,10)(7,12,22,16)(8,17)(9,21)(14,30,20,28)(18,25)(23,29,24,31)(26,32);(1,3)(2,7,9,23,10,22,19,24)(4,16)(5,20)(6,13)(8,11,25,28,26,27,15,30)(12,18)(14,21)(17,29)(31,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,e^c=eh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1531);